DP Mathematics: Applications and Interpretation Questionbank

The graph of a function $f$ passes through the point $\left(\ln 4, 20\right)$.

Given that $f\text{'}\left(x\right)=6{\text{e}}^{2x}$, find $f\left(x\right)$.

Show that $f\text{'}\text{'}\left(x\right)=\frac{24-6x^2}{{\left(x^2+4\right)}^2}$.

Find the least value of $n$.

Find $\int \frac{6x}{x^2+4}\text{d}x$.

Let $R$ be the region enclosed by the graph of $f$, the $x$-axis and the lines $x=1$ and $x=3$. The area of $R$ is $19.6$, correct to three significant figures.

Find $f\left(x\right)$.

Find ${\int }_0^5\frac{1000}{2+t}\text{d}t$.

State in context what this value represents.

Determine ${\int }_0^{365}P\left(t\right) \text{d}t$ and state what it represents.

Find an expression for $P$ in terms of $t$.

Expand ${\left(\frac{1}{u}+1\right)}^2$.

Find $\int {\left(\frac{1}{\left(x+2\right)}+1\right)}^{2}dx$.

A function $f$ satisfies the conditions $f\left(0\right)=-4$, $f\left(1\right)=0$ and its second derivative is $f^{″}\left(x\right)=15\sqrt{x}+\frac{1}{{\left(x+1\right)}^2}$, $x$ ≥ 0.

Find $f\left(x\right)$.

Write down the $x$-coordinate of the point of inflexion on the graph of $y=f\left(x\right)$.

find the value of $f\left(1\right)$.

find the value of $f\left(4\right)$.

Sketch the curve $y=f\left(x\right)$, 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

Find $f^{\prime }\left(x\right)$.

Find $g^{\prime }\left(x\right)$.

Hence, or otherwise, find $\underset{0}{\overset{\pi }{\int }}{\text{e}}^{-x}\text{sin}x\text{d}x$.

Find $\int \left(6x^2-3x\right)\text{d}x$.

Find the area of the region enclosed by the graph of $f$, the x-axis and the lines x = 1 and x = 2 .

Find $\int x{\text{e}}^{x^2-1}\text{d}x$.

Find $f(x)$, given that $f^{\prime }(x)=x{\text{e}}^{x^2-1}$ and $f(-1)=3$.

Let $f^{\prime }(x)=\frac{3x^2}{{(x^3+1)}^5}$. Given that $f(0)=1$, find $f(x)$.

Find $\int {\left(f\left(x\right)\right)}^2\text{d}x$.

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 . Find the volume of the solid formed when R is revolved 360° about the x-axis.

Let $f\left(x\right)=\frac{6-2x}{\sqrt{16+6x-x^2}}$. The following diagram shows part of the graph of $f$.

The region R is enclosed by the graph of $f$, the $x$-axis, and the $y$-axis. Find the area of R.

Let $f^{\prime }(x)={\sin }^3(2x)\cos (2x)$. Find $f(x)$, given that $f\left(\frac{\pi }{4}\right)=1$.

The derivative of a function $f$ is given by $f^{\prime }(x)=2{\text{e}}^{-3x}$. The graph of $f$ passes through $\left(\frac{1}{3}\text{,}5\right)$.

Find $f(x)$.

Find $\frac{\text{d}y}{\text{d}x}$.

Hence find $\int \left(3x^2+1\right)\sqrt{x^3+x}\text{d}x$.

Write down an expression for the area of $R$.

Hence find the exact area of $R$.

${\int }_{-2}^0\left(f\left(x\right)\text{ + 2}\right)\text{d}x$.

${\int }_{-2}^{0}f\left(x\text{ + 2}\right)\text{d}x$.